Harmonically balanced capitulation over quadratic fields of type (9,9)

TL;DR

This paper investigates the structure of Galois groups of 3-class field towers over quadratic fields with specific class group types, using Artin patterns and computational group theory to identify new phenomena and bounds.

Contribution

It introduces a method to determine Galois group types via Artin patterns and extends the SmallGroups database to analyze groups with harmonic transfer kernels.

Findings

Identification of new group phenomena with harmonic transfer kernels

Bounds for the relation rank depending on quadratic field signatures

Extension of the SmallGroups database for larger p-groups

Abstract

The isomorphism type of the Galois group G of finite 3-class field towers of quadratic number fields with 3-class group of type (9,9) is determined by means of Artin patterns which contain information on the transfer of 3-classes to unramified abelian 3-extensions. First, as an approximation of the group G, its metabelianization M=G/G", which is isomorphic to the Galois group of the second Hilbert 3-class field, is sought by sifting the SmallGroups library with the aid of pattern recognition. In cases with order |M|>3^8, the SmallGroups database must be extended by means of the p-group generation algorithm, which reveals new phenomena of groups with harmonically balanced transfer kernels and trees with periodic trifurcations. Bounds for the relation rank d2(M) of M in dependence on the signature of the quadratic base field admit the decision whether the derived length of G is dl(G)=2 or dl(G)>=3.

Tables

Table 1. Table 1. Layers of subgroups and field extensions

Layer	Subgroup index resp. extension degree	Cardinality
	$(G:S)=[N:K]=$	$p=2$	$p=3$
${\mathrm{Lyr}}_1$	$p$	$3$	$4$
${\mathrm{Lyr}}_2$	$p^2$	$7$	$13$
${\mathrm{Lyr}}_3$	$p^3$	$3$	$4$
${\mathrm{Lyr}}_4$	$p^4$	$1$	$1$

Table 2. Table 2. Step size- 3 3 3 children of R = ⟨ 243 , 2 ⟩ 𝑅 243 2 R=\langle 243,2\rangle

$i$	SmallGroup	$\nu$	$\mu$	${(N_i,C_i)}_{1\le i\le \nu }$
$22$	$⟨6561,23⟩$	$3$	$5$	$(10,0;22,7;9,9)$
$23$	$⟨6561,24⟩$	$2$	$4$	$(2,0;3,0)$
$24$	$⟨6561,25⟩$	$3$	$5$	$(7,0;14,5;6,6)$
$25$	$⟨6561,26⟩$	$2$	$4$	$(4,0;3,0)$
$26$	$⟨6561,27⟩$	$2$	$4$	$(4,0;3,0)$
$27$	$⟨6561,28⟩$	$2$	$4$	$(3,0;5,0)$

Equations

$$H_1:=⟨x,J_0⟩,H_2:=⟨xy,J_0⟩,H_3:=⟨x{y}^2,J_0⟩,H_4:=⟨y,J_0⟩,$$

$$H_1=⟨x,y^3,G^{\prime }⟩,H_2=⟨xy,y^3,G^{\prime }⟩,H_3=⟨x{y}^2,y^3,G^{\prime }⟩,H_4=⟨y,x^3,G^{\prime }⟩,$$

$$\text{three subgroups of }{H}_1,J_{11}$$

$$\text{three subgroups of }{H}_2,J_{21}$$

$$\text{three subgroups of }{H}_3,J_{31}$$

$$\text{three subgroups of }{H}_4,J_{41}$$

$$K_1:=⟨x^3,G^{\prime }⟩,K_2:=⟨x^3{y}^3,G^{\prime }⟩,K_3:=⟨x^3{y}^6,G^{\prime }⟩,K_4:=⟨y^3,G^{\prime }⟩,$$

$$d_1\le d_2\le d_1+r+\theta ,$$

$$\theta =\{\begin{array}{ll}1& \text{if }K\text{ contains a primitive }p\text{-th root of unity},\\ 0& \text{otherwise}.\end{array}$$

$$2\le d_2\le 2+r,$$

$$r=\{\begin{array}{ll}1& \text{for real }K\text{ with }(r_1,r_2)=(2,0),\\ 0& \text{for imaginary }K\text{ with }(r_1,r_2)=(0,1).\end{array}$$

$$T_{i+1,j}/T_{i+1,j}^{\prime \prime }\simeq \{\begin{array}{ll}{S}_{i+1}& \text{ if }i=0,\\ M& \text{ if }i\ge 1,\end{array}$$

$$(\forall 1\le i\le n)\ker (T_{G,S_i})=S_{\pi (i)}/G^{\prime },\text{ abbreviated by }\ker (T_{G,S_i})=S_{\pi (i)}.$$

$${\tau }_0$$

$${\varkappa }_2$$

$${\tau }_0$$

$${\tau }_2$$

$${\varkappa }_2$$

$${\tau }_0$$

$${\tau }_2$$

$${\varkappa }_2$$

$${\tau }_0$$

$${\tau }_2$$

$${\varkappa }_2$$

$${\tau }_0$$

$${\tau }_2$$

$${\varkappa }_2$$

$${\tau }_0$$

$${\tau }_2$$

$${\varkappa }_2$$

$${\tau }_0$$

$${\tau }_2$$

$${\varkappa }_2$$

$$\bigl{(}\tau(J_{11}),\tau(J_{12}),\tau(J_{13}),\tau(J_{21}),\tau(J_{22}),\tau(J_{23}),\tau(J_{31}),\tau(J_{32}),\tau(J_{33}),\tau(J_{41}),\tau(J_{42}),\tau(J_{43});\tau(J_{0})\bigr{)}$$

$$\tau_{2,1}=\bigl{(}[(411)^{3},321]^{12};[(321)^{12},331]\bigr{)},$$

$$\tau_{2,1}=\bigl{(}[(511)^{3},331]^{12};[(322)^{4},(331)^{9}]\bigr{)}.$$

$${\tau }_0$$

Full text

Research supported by the Austrian Science Fund (FWF): project P 26008-N25

1. Introduction

The investigation of $3$-class field towers of quadratic fields $K$ with $3$-class group $\mathrm{Cl}_{3}(K)\simeq C_{9}\times C_{9}$ was suggested by Arnold Scholz in a letter of 12 October 1930 to Helmut Hasse [11, pp. 155–156]. Due to limited computational facilities at that time, Scholz could not find suitable base fields $K$, and consequently his project remained unfinished up to now. In collaboration with M. F. Newman [22], we intentionally prepared the revival of the Scholz project by studying the analogous case of $2$-class field towers of quadratic fields $K$ with $2$-class group $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$, inspired by research results of Benjamin and Snyder [4]. On the one hand, the $2$-power extensions are computationally easier because of their modest degrees, but on the other hand, the broader variability of their transfer kernels discourages the occurrence of harmonically balanced capitulation (§ 7). A common feature of all these $p$-class groups $G=\mathrm{Gal}(\mathrm{F}_{p}^{1}(K)/K)\simeq\mathrm{Cl}_{p}(K)\simeq C_{p^{2}}\times C_{p^{2}}$ with $p\in\{2,3\}$ is their consistence of four layers of intermediate subgroups $1\leq S<G$ corresponding to four layers of unramified abelian $p$-extensions $K<N\leq\mathrm{F}_{p}^{1}(K)$ within the Hilbert $p$-class field $\mathrm{F}_{p}^{1}(K)$ of $K$, according to the Artin reciprocity law [2] and the Galois correspondence. For a coarse overview with cardinalities see Table 1, for finer details about generators see § 3, in particular Figure 1.

Similarly as in [22], we shall use the strategy of pattern recognition via Artin transfers: we show that computational arithmetical information on the transfer $T_{K,N}:\,\mathrm{Cl}_{3}(K)\to\mathrm{Cl}_{3}(N)$ of $3$-classes from the base field $K$ to the fields $N$ with relative degrees $3$ and $9$ in the first and second layer suffices for determining the metabelianization $M=\mathfrak{G}/\mathfrak{G}^{\prime\prime}$ of the $3$-class field tower group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ of $K$, i.e. the Galois group of the maximal unramified pro-$3$ extension $\mathrm{F}_{3}^{\infty}(K)$ of $K$. Data for fields with degrees $27$ and $81$ in the third and fourth layer can be derived by purely group theoretic methods from a presentation of $M$ without the need of any number theoretic calculations. In particular, the structure of the $3$-class group $\mathrm{Cl}_{3}(\mathrm{F}_{3}^{1}(K))$ of the first Hilbert $3$-class field $\mathrm{F}_{3}^{1}(K)$ of $K$, which alone forms the fourth layer, can be obtained in this way.

2. Arithmetic and algebraic Artin pattern

The input for the process of pattern recognition consists of arithmetic information about an assigned number field, collected in its Artin pattern.

Definition 2.1**.**

The number theoretic Artin pattern $\mathrm{AP}(K)$ of an algebraic number field $K$ with respect to a prime $p$ is defined as the collection $(\varkappa(K),\tau(K))$ of all kernels $\varkappa(K)=(\ker(T_{K,N}))_{N}$ and targets $\tau(K)=(\mathrm{Cl}_{p}(N))_{N}$ of extension homomorphisms $T_{K,N}:\,\mathrm{Cl}_{p}(K)\to\mathrm{Cl}_{p}(N)$ of $p$-class groups from $K$ to unramified abelian extensions $K\leq N\leq\mathrm{F}_{p}^{1}(K)$ of $K$ within the first Hilbert $p$-class field $\mathrm{F}_{p}^{1}(K)$ of $K$.

The idea of the number theoretic (arithmetic) Artin pattern $\mathrm{AP}(K)=(\varkappa(K),\tau(K))$ was first introduced in [15], using the terminology transfer kernel type (TKT) $\varkappa(K)$ and transfer target type (TTT) $\tau(K)$. It is determined by means of class field theoretic routines of the computational algebra system Magma [6, 7, 12].

Before the arithmetic information can be exploited as a search pattern for a database query in the SmallGroups Library [5] it must be interpreted as algebraic information, by means of the Artin reciprocity law of class field theory [2, 3].

Definition 2.2**.**

Given a prime $p$, the group theoretic Artin pattern $\mathrm{AP}(G)$ of a pro-$p$ group $G$ consists of all kernels $\varkappa(G)=(\ker(T_{G,S}))_{S}$ and targets $\tau(G)=(S/S^{\prime})_{S}$ of Artin transfer homomorphisms $T_{G,S}:\,G/G^{\prime}\to S/S^{\prime}$ from $G$ to normal subgroups $G^{\prime}\leq S\leq G$ of finite index $(G:S)<\infty$ containing the commutator subgroup $G^{\prime}$ of $G$.

For the process of pattern recognition, the group $G$ is taken as the metabelian Galois group $M=\mathrm{Gal}(\mathrm{F}_{p}^{2}(K)/K)$ of the second Hilbert $p$-class field $\mathrm{F}_{p}^{2}(K)$ of a number field $K$, briefly called the second $p$-class group of $K$ [14]. The following result admits the identification of arithmetic and algebraic data.

Theorem 2.1**.**

Let $p$ be a prime, then the group theoretic Artin pattern $\mathrm{AP}(M)$ of the second $p$-class group $M=\mathrm{Gal}(\mathrm{F}_{p}^{2}(K)/K)$ of $K$ coincides with the number theoretic Artin pattern $\mathrm{AP}(K)$ of $K$, and the commutator subgroup $M^{\prime}$ of $M$ is isomorphic to the $p$-class group $\mathrm{Cl}_{p}(\mathrm{F}_{p}^{1}(K))\simeq\mathrm{Gal}(\mathrm{F}_{p}^{2}(K)/\mathrm{F}_{p}^{1}(K))$ of the first Hilbert $p$-class field $\mathrm{F}_{p}^{1}(K)$ of $K$.

Proof.

See [18, § 1.6, pp. 77–79], in particular [18, Thm. 1.12, p. 77]. ∎

3. Normal lattice and layers of $G/G^{\prime}\simeq C_{9}\times C_{9}$

Let $G$ be a finite $3$-group with two generators $x,y$ such that $G=\langle x,y\rangle$ and $G/G^{\prime}\simeq C_{9}\times C_{9}$, that is, $x^{9},y^{9}\in G^{\prime}$ but $x^{3},y^{3}\notin G^{\prime}$. Here $G^{\prime}$ denotes the derived subgroup of $G$. Let $\mathrm{Lyr}_{0}(G)=\{G\}$ be the 0th layer with $\tau_{0}=(G/G^{\prime})$. In the following description of the normal lattice of $G$ above $G^{\prime}$, we give generators for the members of each layer mentioned in Table 1.

Proposition 3.1**.**

The Frattini subgroup $J_{0}:=\Phi(G)=\langle x^{3},y^{3},G^{\prime}\rangle$ of $G$ is the intersection of the four maximal subgroups which form the first layer $\mathrm{Lyr}_{1}(G)$:

[TABLE]

with Artin pattern $(\varkappa_{1},\tau_{1})$. They are of index $3$ in $G$ and can also be viewed as

[TABLE]

which shows that $H_{i}/G^{\prime}\simeq C_{9}\times C_{3}$, for each $1\leq i\leq 4$, whereas $J_{0}/G^{\prime}\simeq C_{3}\times C_{3}$.

In addition to $J_{0}$, there are twelve subgroups $J_{ik}$ with $1\leq i\leq 4$, $1\leq k\leq 3$, of index $9$ in $G$ which belong to the second layer $\mathrm{Lyr}_{2}(G)$:

[TABLE]

with Artin pattern $(\varkappa_{2},\tau_{2})$. They all have a cyclic quotient $J_{ik}/G^{\prime}\simeq C_{9}$.

Finally, there are four subgroups of index $27$ in $G$ which form the third layer $\mathrm{Lyr}_{3}(G)$:

[TABLE]

with Artin pattern $(\varkappa_{3},\tau_{3})$. They have quotient $K_{i}/G^{\prime}\simeq C_{3}$, for each $1\leq i\leq 4$.

Additionally, let $\mathrm{Lyr}_{4}(G)=\{G^{\prime}\}$ be the fourth layer with $\tau_{4}=(G^{\prime}/G^{\prime\prime})$.

Logarithmic abelian type invariants are used for the components of $\tau$, for instance, $(311)$ or even $(31^{2})$, with formal exponents indicating iteration, instead of $(27,3,3)$. See [20, § 2.1, Eqn. (2.3), p. 661].

Figure 1 shows that the normal lattice of $G$ above the derived subgroup $G^{\prime}$ consists of four similar parts ($1\leq i\leq 4$), agglutinated at the subgroups $G$, $J_{0}=\Phi(G)$ and $G^{\prime}$.

4. Cohomology criterion and Schur $\sigma$-groups

Let $p$ be a prime number and $G$ be a pro-$p$ group. Then the finite field $\mathbb{F}_{p}$ is a trivial $G$-module, and two crucial cohomological invariants of $G$ are the generator rank $d_{1}:=d_{p,1}(G):=\dim_{\mathbb{F}_{p}}\mathrm{H}^{1}(G,\mathbb{F}_{p})$ and the relation rank $d_{2}:=d_{p,2}(G):=\dim_{\mathbb{F}_{p}}\mathrm{H}^{2}(G,\mathbb{F}_{p})$. Shafarevich has established the following necessary criterion.

Theorem 4.1**.**

The Galois group $G=\mathrm{Gal}(\mathrm{F}_{p}^{\infty}(K)/K)$ of the Hilbert $p$-class field tower $\mathrm{F}_{p}^{\infty}(K)$ of a number field $K$, that is the maximal unramified pro-$p$ extension of $K$, is confined to constraints in terms of the generator rank and relation rank of $G$:

[TABLE]

where $r$ and $\theta$ are defined in the following way: if $(r_{1},r_{2})$ denotes the signature of $K$, then the torsion free Dirichlet unit rank of $K$ is given by $r=r_{1}+r_{2}-1$ and

[TABLE]

Proof.

See [16, Thm. 5.1, p. 28]. ∎

Corollary 4.1**.**

Applied to the particular case of $p=3$ and the $3$-class field tower group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ of a quadratic number field $K$ with $3$-class group $\mathfrak{G}/\mathfrak{G}^{\prime}\simeq\mathrm{Cl}_{3}(K)\simeq C_{9}\times C_{9}$, and thus with Frattini quotient $\mathfrak{G}/\Phi(\mathfrak{G})\simeq C_{3}\times C_{3}$, the Shafarevich criterion becomes

[TABLE]

where

[TABLE]

Proof.

The generator rank of $\mathfrak{G}$ is $d_{1}=2$, according to the Burnside basis theorem, $\theta=0$ because the cyclotomic quadratic field $\mathbb{Q}(\sqrt{-3})$ has class number $h=1$, and the Dirichlet unit rank $r$ of $K$ is given in the indicated way. ∎

For imaginary quadratic base fields, the cohomology criterion is sharp, $d_{2}=d_{1}=2$, and motivates the following definitions.

Definition 4.1**.**

The pro-$p$ group $G$ is called a Schur group if it has a balanced presentation with coinciding relation rank and generator rank, $d_{2}=d_{1}$.

If there exists an automorphism $\sigma\in\mathrm{Aut}(G)$ which acts as inversion on $\mathrm{H}^{1}(G,\mathbb{F}_{p})$, that is $(\forall\,x\in\mathrm{H}^{1}(G,\mathbb{F}_{p}))\,x^{\sigma}=x^{-1}$, then $G$ is called a $\sigma 1$-group or briefly $\sigma$-group.

If $\sigma$ additionally acts as inversion on $\mathrm{H}^{2}(G,\mathbb{F}_{p})$, then $G$ is called a $\sigma 2$-group.

Theorem 4.2**.**

For an odd prime $p\geq 3$, the $p$-class field tower group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{p}^{\infty}(K)/K)$ of an imaginary quadratic field must be a Schur $\sigma$-group. More generally, the $p$-class field tower group $\mathfrak{G}$ and all higher $p$-class groups $\mathrm{Gal}(\mathrm{F}_{p}^{n}(K)/K)$, $n\geq 2$, must be $\sigma 2$-groups.

Proof.

The first result is due to Koch and Venkov [10]. See also the papers by Arrigoni [1] and Boston, Bush, Hajir [8], where a $\sigma$-group is called a group with generator inverting (GI) automorphism. The second claim was proved by Schoof [26]. See also Boston, Bush, Hajir [9], and our paper [21, Dfn. 4.3, p. 83], where a $\sigma 2$-group is called a group with relator inverting (RI) automorphism. ∎

5. Metabelian root of type $(9,9)$

A crucial non-abelian root of finite $3$-groups $G$ with derived quotient $G/G^{\prime}\simeq C_{9}\times C_{9}$ is the metabelian $3$-group $R:=\langle 243,2\rangle$ of order $243=3^{5}$.

Proposition 5.1**.**

$R$* is a highly capable $\sigma$-group with nuclear rank $\nu(R)=5$, $p$-multiplicator rank $\mu(R)=5$ and descendant numbers $(N_{i},C_{i})_{1\leq i\leq 5}=(11,2;58,32;58,58;11,11;1,1)$. Invariants and SmallGroup identifiers [13] for several crucial immediate descendants of step size $s=3$ of $R$ are given in Table 2.*

With respect to imaginary quadratic fields of type $(9,9)$, the immediate descendants $R-\#3;i$ with step size $s=3$ of $R$, given in Proposition 5.1, are most important. Their locations in the descendant tree of $R$ are drawn in Figure 2.

6. Periodic trifurcations

Theorem 6.1**.**

*(Tree with periodic trifurcations and parametrized sequences of Schur $\sigma$-groups)

For any upper bound $u\leq 30$, there exists a sequence $(V_{i})_{0\leq i\leq u}$ of $\sigma$-groups with nuclear rank $\nu(V_{i})=3$ such that $(\forall\,0\leq i<u)$*

1. (1)

$V_{i}=\pi_{3}(V_{i+1})$* is parent of $V_{i+1}$ with step size $s=3$, $V_{i+1}$ is the unique $\sigma$-child of $V_{i}$ with step size $s=3$ and nuclear rank $\nu(V_{i+1})=3$, and* 2. (2)

$V_{i}$* possesses a unique $\sigma$-child $S_{i+1}$ with step size $s=3$, $V_{i}=\pi_{3}(S_{i+1})$, with nuclear rank $\nu(S_{i+1})=2$ and descendant numbers $(N_{2},C_{2})=(3,0)$. $S_{i+1}$ is a sibling of $V_{i+1}$.*

All three immediate descendants $T_{i+1,j}$, $1\leq j\leq 3$, of $S_{i+1}$ with step size $s=2$, $S_{i+1}=\pi_{2}(T_{i+1,j})$, are non-metabelian Schur $\sigma$-groups $T_{i+1,j}=S_{i+1}-\#2;j$, $1\leq j\leq 3$.

The minimal possible selection of the root is either $V_{0}=\langle 243,2\rangle-\#3;22=\langle 6561,23\rangle$ or $V_{0}=\langle 243,2\rangle-\#3;24=\langle 6561,25\rangle$.

For this minimal selection, the metabelianization of the non-metabelian Schur $\sigma$-groups $T_{i+1,j}=S_{i+1}-\#2;j$, $1\leq j\leq 3$, is given by

[TABLE]

for a fixed step size-$1$ child $M$ of $V_{1}$.

The metabelianization $M$ is given by $V_{1}-\#1;4$ for the selection $V_{0}=\langle 6561,23\rangle$, resp. $V_{1}-\#1;3$ for the selection $V_{0}=\langle 6561,25\rangle$. See Figure 3.

Proof.

All claims have been verified with the aid of Magma [12] up to $u=30$ and logarithmic order $100$, that is, for all descendants of orders not exceeding $3^{100}$. ∎

Corollary 6.1**.**

$S_{1}$* is the metabelianization with child topology of $T_{1,j}$ with $1\leq j\leq 3$, and $M$ is the common metabelianization with fork topology of all $T_{i,j}$ with $i\geq 2$ and $1\leq j\leq 3$. The common fork for all $i\geq 2$ is $V_{1}$. (These concepts have been introduced in [18, § 5, pp. 89–92].)*

Conjecture 6.1**.**

Theorem 6.1 remains true for any $u>30$.

In these descendant trees, derived length $\mathrm{dl}=3$ sets in partially with order $3^{12}$, completely with order $3^{13}$, $\mathrm{dl}=4$ partially with order $3^{27}$, completely with order $3^{29}$, and $\mathrm{dl}=5$ partially with order $3^{60}$, completely with order $3^{61}$.

7. Capitulation laws

With $p$ a prime number, let $G$ be a finite $p$-group.

Definition 7.1**.**

Let $\{S_{1},\ldots,S_{n}\}$ be the set of intermediate subgroups $G^{\prime}\leq S\leq G$ between $G$ and its derived subgroup $G^{\prime}$, let $T_{G,S_{i}}:\,G/G^{\prime}\to S_{i}/S_{i}^{\prime}$ be the collection of Artin transfers from $G$ to $S_{i}$, $1\leq i\leq n$, and denote by $\mathfrak{S}_{n}$ the symmetric group on $n$ letters. The group $G$ is said to possess harmonically balanced capitulation if there exists some permutation $\pi\in\mathfrak{S}_{n}$ such that

[TABLE]

Remark 7.1**.**

We shall restrict our attention to groups $G$ with arithmetical capitulation satisfying the following two axioms $(\forall S\leq G)$ ($\ker(T_{G,S})=G^{\prime}$ $\Longrightarrow$ $S=G$) and $\ker(T_{G,G^{\prime}})=G$, motivated by Hilbert’s Theorem 94 and the Artin-Furtwängler principal ideal theorem.

Example 7.1**.**

In [14] and [15], we thoroughly investigated the capitulation over quadratic fields $K$ with elementary bicyclic $p$-class group $\mathrm{Cl}_{p}(K)\simeq C_{p}\times C_{p}$ for $p=3$. In [18], we looked at the analogue for $p\in\{5,7\}$. Since arithmetical capitulation requires the mandatory transposition $G\leftrightarrow G^{\prime}$, harmonically balanced capitulation can be described by a permutation $\pi\in\mathfrak{S}_{p+1}$ of the $p+1$ proper intermediate subgroups $G^{\prime}<S_{1},\ldots,S_{p+1}<G$.

For $p=3$, only two permutations $\pi\in\mathfrak{S}_{4}$ are admissible. They were called capitulation type

$\mathrm{G}.16$, $\varkappa=(1243)$ (two fixed points and a transposition), and

$\mathrm{G}.19$, $\varkappa=(2143)$ (two disjoint transpositions), [14, Tbl. 3–4, pp. 497–498],

and they enforce a $3$-class field tower with at least three stages, $\ell_{3}(K)\geq 3$ [18, Tbl. 1, p. 90].

For $p=5$, however, a lot of permutations $\pi\in\mathfrak{S}_{6}$ can occur [18, Tbl. 3, p. 94], for instance five cases which admit a metabelian $5$-class field tower with two stages [18, Thm. 7.4, pp. 96-97], the identity with six fixed points, a $4$-cycle with two fixed points, a $5$-cycle with a single fixed point, two disjoint $3$-cycles, and a $6$-cycles. The latter two permutations are fixed point free, just as type $\mathrm{G}.19$ for $p=3$, which is described by Scholz with the following allegory: “So wie es sich für wohlerzogene Bürger geziemt: jeder läßt einem anderen den Vortritt aber jeder kommt auch selbst heran”[11, pp. 155–156].

Similarly, for $p=7$ there are also many admissible permutations $\pi\in\mathfrak{S}_{8}$ [18, Tbl. 4, p. 95].

Not every permutation $\pi\in\mathfrak{S}_{n}$ will be admissible in Formula (7.1), because there exist some general laws which have to be obeyed by the capitulation. We state these laws for the special case of the normal lattice in § 3. So let $G$ be a finite $3$-group with non-elementary bihomocyclic $G/G^{\prime}\simeq C_{9}\times C_{9}$.

Proposition 7.1**.**

The capitulation kernels satisfy the following inclusions:

1. (1)

$(\forall 1\leq i\leq 4)\,(\forall 1\leq k\leq 3)\quad G^{\prime}<\ker(T_{G,H_{i}})\leq\ker(T_{G,J_{ik}})\leq\ker(T_{G,K_{i}})\leq G$, 2. (2)

$(\forall 1\leq i,j\leq 4)\quad G^{\prime}<\ker(T_{G,H_{i}})\leq\ker(T_{G,J_{0}})\leq\ker(T_{G,K_{j}})\leq G$.

Proof.

For nested subgroups $K<H<G$, we generally have the compositum $T_{G,K}=T_{H,K}\circ T_{G,H}$ of Artin transfers, and consequently $T_{G,K}(x)=T_{H,K}(T_{G,H}(x))=T_{H,K}(1)=1$ for all $x\in\ker(T_{G,H})$, that is $\ker(T_{G,H})\leq\ker(T_{G,K})$. Further, $\ker(T_{G,G})=G^{\prime}$ and $\ker(T_{G,G^{\prime}})=G$. ∎

Remark 7.2**.**

For imaginary quadratic base fields, three principal scenarios of capitulation seem to occur. The transfer kernels are either bi-polarized, e.g. $\varkappa_{1}=(K_{4},K_{1},K_{1},K_{1})$, and thus necessarily $\ker(T_{G,J_{0}})=J_{0}$, or uni-polarized, e.g. $\varkappa_{1}=(K_{1},K_{1},K_{1},K_{1})$ with exceptional $\ker(T_{G,J_{0}})=J_{11}$, or harmonically balanced (the most frequent scenario), as described by the following proposition.

Proposition 7.2**.**

$G$* possesses harmonically balanced capitulation (briefly $\mathrm{HBC}$) if and only if $\ker(T_{G,J_{0}})=J_{0}$ and there exists a permutation $\lambda\in\mathfrak{S}_{4}$ such that for all $1\leq i\leq 4$*

1. (1)

$\ker(T_{G,H_{i}})=K_{\lambda(i)}$, 2. (2)

$(\exists\rho_{i}\in\mathfrak{S}_{3})$* $(\forall 1\leq k\leq 3)$ $\ker(T_{G,J_{ik}})=J_{\lambda(i),\rho_{i}(k)}$,* 3. (3)

$\ker(T_{G,K_{i}})=H_{\lambda(i)}$.

Proof.

This is a consequence of Definition 7.1 under the constraints of Proposition 7.1. ∎

The five permutations involved in harmonically balanced capitulation are illustrated in Figure 4.

8. Imaginary quadratic fields of type $(9,9)$

Since Scholz and Taussky [25] were generally interested in the way how more and more ideal classes of order a power of $p$ successively become principal in the subfields $N$ of the Hilbert $p$-class field $\mathrm{F}_{p}^{1}(K)$ of a base field $K$ with non-trivial $p$-class group, it might well be true that the genius Scholz intended to investigate $3$-class field towers of imaginary quadratic fields $K$ with $3$-class group $\mathrm{Cl}_{3}(K)\simeq C_{9}\times C_{9}$, because he had a presentiment that the capitulation will frequently be harmonically balanced. In the sequel, we show that this is indeed the case.

Theorem 8.1**.**

(Two-stage tower 1.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)=2$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to one of the two metabelian Schur $\sigma 2$-groups $\langle 6561,28\rangle-\#2;i$ with $i=4$, resp. $i=5$, coclass $7$, $\tau_{3}=((222)^{4})$, $\varkappa_{3}=(H_{1},H_{2},H_{3},H_{4})$ and $\tau_{4}=(222)$. Both groups possess harmonically balanced capitulation: $\lambda=1$, $\rho_{1}=(132)$, $\rho_{2}=\rho_{3}=\rho_{4}=1$, resp. $\rho_{2}=\rho_{3}=(132)$, $\rho_{4}=(123)$.

Example 8.1**.**

The information in Formula (8.1) occurs rather sparsely as Artin pattern of imaginary quadratic fields $K$ with $3$-class group of type $(9,9)$. The absolutely smallest discriminant $d_{K}$ with this pattern is $-426\,291=-3\cdot 142\,097$. Since the ordering of the members of $\varkappa_{2}$ cannot be determined strictly by arithmetic computations, it is impossible to decide whether $i=4$ or $i=5$.

Proof.

(Proof of Theorem 8.1) The descendant tree of the metabelian but non-abelian root $\langle 243,2\rangle$ with depth $2$ and step size up to $3$ is constructed by means of the $p$-group generation algorithm [23, 24], which is implemented in Magma [6, 7, 12]. The resulting $27\,222$ vertices are sifted by pattern recognition with respect to the transfer targets $\tau_{0}=(22)$, $\tau_{1}=((221)^{4})$, $\tau_{2}=((321)^{12};222)$, and the relation rank $d_{2}=2$ of the required balanced presentation, but without taking into consideration the transfer kernels $\varkappa_{1}$ and $\varkappa_{2}$. This filtering process unambiguously leads to the two metabelian Schur $\sigma 2$-groups $\langle 6561,28\rangle-\#2;i$ with $i=4$ resp. $i=5$, which also satisfy the requirements of harmonically balanced capitulation. So the assumptions concerning $\varkappa_{1}$ and $\varkappa_{2}$ could be removed from Theorem 8.1. The other three siblings are also metabelian Schur $\sigma 2$-groups but they possess different second layer Artin patterns:

$\tau_{2}=((222)^{3},(321)^{6},(222)^{3};222)$ for $i=1$,

$\tau_{2}=((222)^{3},(321)^{3},(222)^{6};222)$ for $i=2$,

$\tau_{2}=((222)^{3},(321)^{9};222)$ for $i=3$.

This additional information could be useful for studying other base fields of type $(9,9)$. ∎

Theorem 8.2**.**

(Two-stage tower 2a.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)=2$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to one of the two metabelian Schur $\sigma$-groups $\langle 6561,26\rangle-\#2;i$ with $i=1$ resp. $i=3$, coclass $7$, $\tau_{3}=((321)^{2}),222,321$, $\varkappa_{3}=(H_{4},H_{1},H_{3},H_{2})$ and $\tau_{4}=(222)$. Both groups possess $\mathrm{HBC}$: $\lambda=(142)$, $\rho_{1}=(23)$, $\rho_{2}=\rho_{3}=1$, $\rho_{4}=(13)$, resp. $\rho_{2}=\rho_{3}=(123)$, $\rho_{4}=(23)$.

Theorem 8.3**.**

(Two-stage tower 2b.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)=2$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to one of the two metabelian Schur $\sigma$-groups $\langle 6561,27\rangle-\#2;i$ with $i=1$ resp. $i=2$, coclass $7$, $\tau_{3}=(321,222,(321)^{2})$, $\varkappa_{3}=(H_{4},H_{2},H_{1},H_{3})$ and $\tau_{4}=(222)$. Both groups possess $\mathrm{HBC}$: $\lambda=(143)$, $\rho_{1}=(23)$, $\rho_{2}=\rho_{3}=1$, $\rho_{4}=(23)$, resp. $\rho_{2}=\rho_{3}=(123)$, $\rho_{4}=(12)$.

Proof.

(Simultaneous proof of Theorem 8.2 and Theorem 8.3) Again we start at the metabelian but non-abelian root $\langle 243,2\rangle$ and construct its descendant tree with depth $2$ and step size up to $3$ by means of the $p$-group generation algorithm. Again the result consists of $27\,222$ vertices which are sifted by pattern recognition with respect to the occupation numbers of the unordered transfer targets $\tau_{0}=(22)$, $\tau_{1}=((311)^{3},221)$, $\tau_{2}=((42)^{9},(321)^{3};222)$, which coincide for (8.2) and (8.3), and the relation rank $d_{2}=2$ of the required balanced presentation, but without taking into consideration the transfer kernels $\varkappa_{1}$ and $\varkappa_{2}$. This filtering process leads to precisely four metabelian Schur $\sigma 2$-groups, $\langle 6561,26\rangle-\#2;i$ with $i=1$ resp. $i=3$, and $\langle 6561,27\rangle-\#2;i$ with $i=1$ resp. $i=2$. They also satisfy the requirements of harmonically balanced capitulation, so again the assumptions concerning $\varkappa_{1}$ and $\varkappa_{2}$ could be removed from Theorems 8.2 and 8.3. For possible later use we note that the other sibling $\langle 6561,26\rangle-\#2;2$, resp. $\langle 6561,27\rangle-\#2;3$, is also a metabelian Schur $\sigma 2$-group with different second layer Artin pattern: $\tau_{2}=((42)^{6},(222)^{3},(42)^{3};222)$, resp. $\tau_{2}=((42)^{3},(222)^{3},(42)^{6};222)$. ∎

Example 8.2**.**

The information in Formula (8.2) resp. (8.3) occurs frequently as Artin pattern of imaginary quadratic fields $K$ with $3$-class group of type $(9,9)$. The absolutely smallest five discriminants $d_{K}$ with these patterns are $-647\,359$ prime, $-654\,468=-2^{2}\cdot 3\cdot 54\,539$, $-682\,811$ prime, $-1\,333\,159=-29\cdot 45\,971$ and $-1\,443\,567=-3\cdot 79\cdot 6091$. Since the terms of $(\varkappa_{1},\tau_{1})$ and $(\varkappa_{2},\tau_{2})$ are produced in random ordering by the class field theoretic routines of Magma [12], it is not possible to decide which of the four contestant groups in Theorems 8.2 and 8.3 actually occurs as $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$.

Theorem 8.4**.**

(Two-stage tower 3a.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)=2$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to the unique metabelian Schur $\sigma$-group $\langle 6561,26\rangle-\#2;2$ with $\tau_{3}=((321)^{2},222,321)$, $\varkappa_{3}=(H_{4},H_{1},H_{3},H_{2})$,$\tau_{4}=(222)$ and coclass $7$. The group possesses harmonically balanced capitulation: $\lambda=(142)$, $\rho_{1}=(23)$, $\rho_{2}=\rho_{3}=(132)$, $\rho_{4}=(12)$.

Theorem 8.5**.**

(Two-stage tower 3b.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)=2$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to the unique metabelian Schur $\sigma$-group $\langle 6561,27\rangle-\#2;3$ with $\tau_{3}=(321,222,(321)^{2})$, $\varkappa_{3}=(H_{4},H_{2},H_{1},H_{3})$, $\tau_{4}=(222)$ and coclass $7$. The group possesses harmonically balanced capitulation: $\lambda=(143)$, $\rho_{1}=(23)$, $\rho_{2}=\rho_{3}=(132)$, $\rho_{4}=(13)$.

Proof.

(Simultaneous proof of Theorem 8.4 and Theorem 8.5) The claims are consequences of the remarks at the end of the simultaneous proof of Theorem 8.2 and Theorem 8.3. ∎

Example 8.3**.**

The information in Formula (8.4) resp. (8.5) occurs rather sparsely as Artin pattern of imaginary quadratic fields $K$ with $3$-class group of type $(9,9)$. The absolutely smallest discriminant $d_{K}$ with this pattern is $-1\,287\,544=-2^{3}\cdot 227\cdot 709$. Since the terms of $(\varkappa_{1},\tau_{1})$ and $(\varkappa_{2},\tau_{2})$ are produced in random ordering by the class field theoretic routines of Magma [12], it is not possible to decide which of the two contestant groups in Theorems 8.4 and 8.5 actually occurs as $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$.

All the preceding theorems in this section have caused the impression that harmonically balanced capitulation is connected with two-stage towers of $3$-class fields, i.e. metabelian towers. Figure 2 shows that the involved $3$-class tower groups $\mathfrak{G}$ are descendants of the finitely capable groups $\langle 6561,i\rangle$ with $26\leq i\leq 28$. The following theorems, however, will supplement the belated motivation why we thoroughly prepared the study of tree topologies for the description of non-metabelian towers in § 6. Now the descendants of the groups $\langle 6561,i\rangle$ with $23\leq i\leq 25$, which also possess harmonically balanced capitulation, will come into the play. Except for $i=24$, they are infinitely capable and give rise to periodic trifurcations and parametrized sequences of Schur $\sigma$-groups, as illustrated in Figure 3.

Theorem 8.6**.**

(Three-stage tower 1.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)\geq 3$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to one of the infinitely many non-metabelian Schur $\sigma$-groups $T_{i,k}$ with $i\geq 1$, $1\leq k\leq 3$, in the descendant tree of $V_{0}=\langle 6561,25\rangle$ with fixed $\tau_{3}=((331)^{4})$, $\varkappa_{3}=(H_{4},H_{3},H_{2},H_{1})$, and variable $\tau_{4}=(2+i,2,2)$ and coclass $5+3i$. The metabelianization $M=\mathfrak{G}/\mathfrak{G}^{\prime\prime}$ of $\mathfrak{G}$ is isomorphic to $S_{1}=V_{0}-\#3;6$, if $i=1$, and to $V_{1}-\#1;3$ with $V_{1}=V_{0}-\#3;3$, if $i\geq 2$. All groups possess harmonically balanced capitulation: $\lambda=(14)(23)$, $\rho_{1}=(23)$, $\rho_{2}=1$, $\rho_{3}=(123)$, $\rho_{4}=(13)$.

Theorem 8.7**.**

(Two-stage tower 4.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)=2$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to one of three metabelian Schur $\sigma$-groups $\langle 6561,24\rangle-\#2;k$ with $1\leq k\leq 3$, $\tau_{3}=((321)^{4})$, $\varkappa_{3}=(H_{4},H_{1},H_{2},H_{3})$, $\tau_{4}=(222)$ and coclass $7$. The groups possess harmonically balanced capitulation: $\lambda=(1432)$, $\rho_{1}=\rho_{4}=1$, $\rho_{2}=(23)$, $\rho_{3}=(12)$.

Example 8.4**.**

The information in Formula (8.6) resp. (8.7) occurs frequently as Artin pattern of imaginary quadratic fields $K$ with $3$-class group of type $(9,9)$. The absolutely smallest seven discriminants $d_{K}$ with these patterns are $-134\,059$ prime, $-298\,483$ prime, $-430\,411$ prime, $-660\,356=-2^{2}\cdot 165\,089$, $-1\,231\,060=-2^{2}\cdot 5\cdot 61\,553$, $-1\,332\,644=-2^{2}\cdot 333\,161$ and $-1\,430\,867=-31\cdot 101\cdot 457$. Since the terms of $(\varkappa_{1},\tau_{1})$ and $(\varkappa_{2},\tau_{2})$ are produced in random ordering by the class field theoretic routines of Magma [12], it is not possible to decide whether one of the infinitely many candidates in Theorem 8.6 or one of the three contestants in Theorem 8.7 actually occurs as $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$, except one uses the following work around with huge computational complexity.

According to [17, Thm. 5.4, pp. 86–87], all higher $p$-class groups $\mathrm{Gal}(\mathrm{F}_{p}^{n}(K)/K)$ with $n\geq 2$ of a number field $K$ share a common Artin pattern $\mathrm{AP}(K)=(\tau(K),\varkappa(K))$, and consequently this collection of invariants is not able to distinguish between different lengths $\ell_{p}(K)\geq 2$ of the $p$-class field tower of $K$. In addition to the (partial) multi-layered Artin pattern $(\tau_{0};\tau_{1},\varkappa_{1};\tau_{2},\varkappa_{2})$, which was used as the preamble of all preceding theorems, higher abelian quotient invariants are now required [18, Dfn. 1.18, p. 78] for distinguishing between towers with two stages and towers with more stages.

Definition 8.1**.**

By the abelian quotient invariants of second order of a pro-$3$ group $G$ with abelianization $G/G^{\prime}\simeq C_{9}\times C_{9}$ we understand the family of transfer targets $\tau_{2,1}:=\tau_{2,1}(G):=$

[TABLE]

with respect to the collections of Artin transfer homomorphisms $(T_{J,S}:\,J/J^{\prime}\to S/S^{\prime})_{S\in\mathrm{Lyr}_{1}(J)}$ for each $J=J_{ik}$, $1\leq i\leq 4$, $1\leq k\leq 3$, and also for $J=J_{0}$.

$\tau_{2,1}$ includes the information $\tau_{3}$ about the subgroups $K_{i}$, $1\leq i\leq 4$, with index $27$ in $G$, but also data for subgroups $S<G$ of index $27$ which do not contain $G^{\prime}$, and which therefore correspond to unramified but non-abelian extensions of relative degree $27$ of $K$, that is of absolute degree $54$. (See also [19, § 7.1, Eqn. (7.1), p. 149] and [20, § 3.2, Eqn. (3.4), p. 664].)

Theorem 8.8**.**

(Stage separation criterion.) Let $K$ be an imaginary quadratic field with Artin pattern (8.6) resp. (8.7). Then the $3$-class field tower of $K$ has length $\ell_{3}(K)=2$ and one of the three possible Galois groups $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ in Theorem 8.7, if and only if

[TABLE]

but it has length $\ell_{3}(K)\geq 3$ and one of the infinitely many possible Galois groups $\mathfrak{G}$ in Theorem 8.6, if

[TABLE]

We conclude this section with a situation, which unambiguously enforces a $3$-class field tower with at least three stages, $\ell_{3}(K)\geq 3$, without the necessity of computing abelian quotient invariants of second order $\tau_{2,1}$ with high computational complexity.

Theorem 8.9**.**

(Three-stage tower 2.) Let $K$ be an imaginary or real quadratic field with $\mathrm{AP}$

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)\geq 3$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to one of the infinitely many non-metabelian Schur $\sigma$-groups $T_{i,k}$ with $i\geq 1$, $1\leq k\leq 3$, in the descendant tree of $V_{0}=\langle 6561,23\rangle$ with fixed $\tau_{3}=(331,322,322,331)$, $\varkappa_{3}=(H_{4},H_{2},H_{3},H_{1})$, and variable $\tau_{4}=(2+i,2,2)$ and coclass $5+3i$. The metabelianization $M=\mathfrak{G}/\mathfrak{G}^{\prime\prime}$ of $\mathfrak{G}$ is isomorphic to $S_{1}=V_{0}-\#3;9$, if $i=1$, and to $V_{1}-\#1;4$ with $V_{1}=V_{0}-\#3;7$, if $i\geq 2$. All groups possess harmonically balanced capitulation: $\lambda=(14)$, $\rho_{1}=(123)$, $\rho_{2}=(13)$, $\rho_{3}=(23)$, $\rho_{4}=1$. The metabelianization $M$ has $d_{2}=4$, forbidden as $\mathfrak{G}$ for any quadratic field $K$.

Example 8.5**.**

The information in Formula (8.8) occurs as Artin pattern of two imaginary quadratic fields and a single real quadratic field $K$ with $3$-class group of type $(9,9)$. The discriminants $d_{K}$ with this pattern are $-475\,355=-5\cdot 95\,071$, $-1\,201\,272=-2^{3}\cdot 3\cdot 50\,053$, and $8\,965\,093=19\cdot 471\,847$. According to Theorem 8.9, these three fields possess a non-metabelian $3$-class field tower of length $\ell_{3}(K)\geq 3$.

Proof.

(Simultaneous proof of Theorem 8.6, Theorem 8.8, Theorem 8.9, and Theorem 8.7) All claims about abelian quotient invariants of second order $\tau_{2,1}$ and about the non-metabelian Schur $\sigma 2$-groups $T_{i,k}$ have been verified with the aid of Magma [12] as described in the proof of Theorem 6.1 and expressed in Conjecture 6.1. The process of pattern recognition was carried out by a similar database query as in the proof of Theorem 8.1, which here resulted in descendants of $\langle 6561,i\rangle$ with $i\in\{23,24,25\}$ instead of $i=28$. ∎

9. Abelian root

The abelian root of finite $3$-groups $G$ with derived quotient $G/G^{\prime}\simeq C_{9}\times C_{9}$ is the non-elementary bihomocyclic $3$-group $A:=\langle 81,2\rangle\simeq C_{9}\times C_{9}$ of order $81=3^{4}$. We expected that the root $A$ and its descendants will not be involved in the investigation of imaginary quadratic fields. Contrary to our assumptions, however, it also seems to be irrelevant for real quadratic fields. All quadratic fields $K$ with $3$-class group $\mathrm{Cl}_{3}(K)\simeq C_{9}\times C_{9}$ have higher non-abelian $3$-class groups $\mathrm{Gal}(\mathrm{F}_{3}^{n}(K)/K)$, $n\geq 2$, in the descendant tree of the non-abelian root $R=\langle 243,2\rangle$ discussed in § 5.

10. Real quadratic fields of type $(9,9)$

Since they start with extremely huge discriminants, only four real quadratic fields $K$ with $3$-class group $\mathrm{Cl}_{3}(K)\simeq C_{9}\times C_{9}$ are known up to now. The discriminants of these fields are

[TABLE]

The construction of the unramified abelian $3$-extensions with relative degree $9$, which form a multiplet with $13$ members (a tridecuplet), consumes much more CPU time for real than for imaginary base fields. Therefore it was hard to analyze the four discriminants completely.

Theorem 10.1**.**

(Two stage tower.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

Then the $3$-class field tower of $K$ has length $\ell_{3}(K)=2$ and Galois group $\mathfrak{G}=\mathrm{Gal}(\mathrm{F}_{3}^{\infty}(K)/K)$ isomorphic to one of the two metabelian $\sigma 2$-groups $\langle 6561,i\rangle$ with $i=383$ resp. $i=384$, coclass $5$, relation rank $d_{2}=3$, $\tau_{3}=(211,(22)^{2},32)$, $\varkappa_{3}=(G,G^{2},H_{1})$ and $\tau_{4}=(22)$. Both groups are far away from possessing harmonically balanced capitulation.

Example 10.1**.**

The information in Formula (10.1) occurs as Artin pattern of three real quadratic fields $K$ with $3$-class group of type $(9,9)$. The discriminants $d_{K}$ with this pattern are $8\,739\,521=7\cdot 1\,248\,503$, $15\,057\,697=43\cdot 350\,179$, and $16\,279\,457=59\cdot 275\,923$. According to Theorem 10.1, these three fields possess a metabelian $3$-class field tower of length $\ell_{3}(K)=2$.

Proof.

(Proof of Theorem 10.1) A similar construction as in the proof of Theorem 8.1 but with looser constraint $d_{2}\leq 3$ for the relation rank yields precisely the two groups in the claim. Here it is sufficient to consider step sizes up to $s=2$ only, instead of $s=3$, whence only $961$ groups are generated, instead of the tedious batch of $27222$. ∎

11. Acknowledgements

The author gratefully acknowledges that his research was supported by the Austrian Science Fund (FWF): project P 26008-N25.